Sometimes it's the drummer, sometimes someone is offstage, sometimes there are entire songs programmed and they all have click tracks in their in-ear monitors, and sometimes it's a combination of those things.
There are so many controllers available now and so many kinds of software that the possibilities are manifold. Drum pads or sensors, foot controllers, even guitar note to midi conversion software can all be used for all kinds of triggers and synchronization.
Many acts have racks of music servers off stage running different effects, synthesizers, and pre-recorded tracks. What's interesting is some of the most synth heavy bands actually create more organic sounding performances. Nine Inch Nails is on record as using Apple Mainstage for software synths and effects while also using totally analog, pieced together custom synth racks and other tools that make their performances really breathe.
If you can dream up a crazy idea for your live show, and you're willing to do the research on what's available and how to use it, and you have a bit (or a lot) of money to throw around, you can do almost anything these days.
Regarding the video linked in the question:
- They all have in-ear monitors and are probably playing with a click
- The drummer has a Roland Octopad trigger pad device
- They have an entire DJ who can literally spend all his time triggering and syncing samples (as necessary) (this is almost definitely where the pre-recorded content is coming from)
- Some "live" music videos actually have additional parts overdubbed after the performance (oh yeah, it's totally "cheating" but whatever makes the money is what happens)
In case it was the drummer, does he just hit a pad and the melody starts playing? How can you be sure the melody will be on tempo?
Three ways (at least) a drummer might trigger something:
- a pad
- a drum trigger attached to a real drum or cymbal
- a foot controller of some kind.
Synchronization can be handled in different ways:
- Musicians can be given a click that matches the backing track, they can just play along with the backing track if they are good enough and the track is clear enough
- The backing track can be made to speed up and slow down to follow the
- The backing can just be sample that doesn't play at a certain tempo
and is repeatedly triggered and plays once for a short time.